## 17Calculus - Path Integrals

##### 17Calculus

Path Integrals

Let's start off with a video. This is a great video which includes a complete introduction, with examples, and explanation of applications of path integrals. It is well worth your time to watch it.

### Dr Chris Tisdell - Integration over curves [45mins-37secs]

video by Dr Chris Tisdell

The equations from the last video are important for evaluating path integrals. Here is a summary.

requirements continuous scalar function $$f(x,y,z)$$ curve $$\mathcal{C}$$ parameterized as $$\vec{c}(t)$$ on an interval $$a \leq t \leq b$$ continuous $$\vec{c}'(t)$$ [derivative of $$\vec{c}(t)$$ with respect to $$t$$]

Most of the above equations should be familiar to you. However, one comment is in order about the term $$f(\vec{c}(t))$$. How do you substitute a vector into a function?
If we write the vector as $$\vec{c}(t) = \langle X(t), Y(t), Z(t) \rangle = X(t) \hat{i} + Y(t) \hat{j} + Z(t)\hat{k}$$ then we substitute the vector components into $$f(x,y,z)$$ as $$x=X(t)$$, $$y=Y(t)$$ and $$z=Z(t)$$. Another way to write this is $$f(x,y,z) = f(X,Y,Z) = f(X(t), Y(t), Z(t))$$. The result is a function of $$t$$ only. All $$x$$'s, $$y$$'s and $$z$$'s will be gone, leaving only an integral in $$t$$.

A very interesting comment he makes in the above video is that the direction of the curve does not affect the answer, i.e. integrating in the opposite direction will give the same answer. This is true because we use only the magnitude of the derivative of the parameterized curve, i.e. $$\| \vec{c}'(t) \|$$, which is the same regardless of the direction. However, the path integral is dependent on the chosen path, i.e. different paths will, in general, give different results. So path integrals are sensitive to the paths.

Okay, let's try some practice problems.

Practice

Unless otherwise instructed, evaluate these path integral over the curve $$\mathcal{C}$$, giving your answers in exact terms.

Unless otherwise instructed, evaluate the path integral $$\int_{\mathcal{C}} x+y ~ds$$ over the curve $$\mathcal{C}:$$ straight line from $$(0,0)$$ to $$(1,1)$$, giving your answer in exact terms.

Problem Statement

Unless otherwise instructed, evaluate the path integral $$\int_{\mathcal{C}} x+y ~ds$$ over the curve $$\mathcal{C}:$$ straight line from $$(0,0)$$ to $$(1,1)$$, giving your answer in exact terms.

Solution

### Dr Chris Tisdell - 858 video solution

video by Dr Chris Tisdell

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, evaluate the path integral $$\int_{\mathcal{C}}{x+y+z ~ ds }$$ over the curve $$\mathcal{C}: \langle \cos(t), \sin(t), t \rangle; 0 \leq t \leq 2\pi$$, giving your answer in exact terms.

Problem Statement

Unless otherwise instructed, evaluate the path integral $$\int_{\mathcal{C}}{x+y+z ~ ds }$$ over the curve $$\mathcal{C}: \langle \cos(t), \sin(t), t \rangle; 0 \leq t \leq 2\pi$$, giving your answer in exact terms.

Solution

### Dr Chris Tisdell - 859 video solution

video by Dr Chris Tisdell

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, evaluate the path integral $$\int_{\mathcal{C}}{x+y+z ~ ds }$$ over the curve $$\mathcal{C}:$$ straight line $$(1,0,0)$$ to $$( 1,0, 2\pi)$$, giving your answer in exact terms.

Problem Statement

Unless otherwise instructed, evaluate the path integral $$\int_{\mathcal{C}}{x+y+z ~ ds }$$ over the curve $$\mathcal{C}:$$ straight line $$(1,0,0)$$ to $$( 1,0, 2\pi)$$, giving your answer in exact terms.

Solution

### Dr Chris Tisdell - 860 video solution

video by Dr Chris Tisdell

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, evaluate the path integral $$f(x,y,z) = x^2+y^2-1+z$$ over the curve $$\mathcal{C}: \langle \cos(t), \sin(t), t \rangle; 0 \leq t \leq 3\pi$$, giving your answer in exact terms.

Problem Statement

Unless otherwise instructed, evaluate the path integral $$f(x,y,z) = x^2+y^2-1+z$$ over the curve $$\mathcal{C}: \langle \cos(t), \sin(t), t \rangle; 0 \leq t \leq 3\pi$$, giving your answer in exact terms.

Solution

### Dr Chris Tisdell - 854 video solution

video by Dr Chris Tisdell

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, evaluate the path integral $$\int_{\mathcal{C}}{ x+y^2 ~ds }$$ over the curve $$\mathcal{C}:$$ 2 lines $$(0,0)$$ - $$(1,1)$$ and $$(1,1)$$ - $$(1,0)$$, giving your answer in exact terms.

Problem Statement

Unless otherwise instructed, evaluate the path integral $$\int_{\mathcal{C}}{ x+y^2 ~ds }$$ over the curve $$\mathcal{C}:$$ 2 lines $$(0,0)$$ - $$(1,1)$$ and $$(1,1)$$ - $$(1,0)$$, giving your answer in exact terms.

Solution

### Dr Chris Tisdell - 861 video solution

video by Dr Chris Tisdell

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, evaluate the path integral $$\int_{\mathcal{C}}{xy^4 ~ds }$$ over the curve $$\mathcal{C}:$$ right half of the circle $$x^2+y^2 = 16$$, giving your answer in exact terms.

Problem Statement

Unless otherwise instructed, evaluate the path integral $$\int_{\mathcal{C}}{xy^4 ~ds }$$ over the curve $$\mathcal{C}:$$ right half of the circle $$x^2+y^2 = 16$$, giving your answer in exact terms.

Solution

### PatrickJMT - 862 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Evaluate $$\int_{\mathcal{C}}{ 2x ~ds }$$ where $$\mathcal{C} = \mathcal{C_1} + \mathcal{C_2}$$; $$\mathcal{C_1}$$ goes from $$(0,0)$$ to $$(1,1)$$ along $$y=x^2$$; $$\mathcal{C_2}$$ goes from $$(1,1)$$ to $$(1,2)$$ along $$x=1$$.

Problem Statement

Evaluate $$\int_{\mathcal{C}}{ 2x ~ds }$$ where $$\mathcal{C} = \mathcal{C_1} + \mathcal{C_2}$$; $$\mathcal{C_1}$$ goes from $$(0,0)$$ to $$(1,1)$$ along $$y=x^2$$; $$\mathcal{C_2}$$ goes from $$(1,1)$$ to $$(1,2)$$ along $$x=1$$.

$$(5\sqrt{5}+11)/6$$

Problem Statement

Evaluate $$\int_{\mathcal{C}}{ 2x ~ds }$$ where $$\mathcal{C} = \mathcal{C_1} + \mathcal{C_2}$$; $$\mathcal{C_1}$$ goes from $$(0,0)$$ to $$(1,1)$$ along $$y=x^2$$; $$\mathcal{C_2}$$ goes from $$(1,1)$$ to $$(1,2)$$ along $$x=1$$.

Solution

### Michel vanBiezen - 2226 video solution

video by Michel vanBiezen

$$(5\sqrt{5}+11)/6$$

Log in to rate this practice problem and to see it's current rating.

Evaluate $$\int_{\mathcal{C}}{x^2z~ds}$$, where $$\mathcal{C}$$ is the line segment from $$(0,6,2)$$ to $$(-1,8,7)$$.

Problem Statement

Evaluate $$\int_{\mathcal{C}}{x^2z~ds}$$, where $$\mathcal{C}$$ is the line segment from $$(0,6,2)$$ to $$(-1,8,7)$$.

Hint

There are an infinite number of ways to parameterize this curve. This instructor uses the equations
$$x=(1-t)x_1+tx_2$$
$$y=(1-t)y_1+ty_2$$
$$z=(1-t)z_1+tz_2$$
$$0 \leq t \leq 1$$.

Problem Statement

Evaluate $$\int_{\mathcal{C}}{x^2z~ds}$$, where $$\mathcal{C}$$ is the line segment from $$(0,6,2)$$ to $$(-1,8,7)$$.

Hint

There are an infinite number of ways to parameterize this curve. This instructor uses the equations
$$x=(1-t)x_1+tx_2$$
$$y=(1-t)y_1+ty_2$$
$$z=(1-t)z_1+tz_2$$
$$0 \leq t \leq 1$$.

Solution

### MIP4U - 2298 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.